Against Phase Veloсities of Elastic Waves in Thin Transversely Isotropic Cylindrical Shell ()
1. Introduction
Based on the use of the dynamic theory of the elasticity for the anisotropic medium and with the help of the hypothesis of thin shells, this paper is determined by the characteristic equation for wave numbers of elastic waves in the thin transversely isotropic cylindrical shell.
2. The Dynamic Theory of the Elasticity for the Transversely Isotropic Medium
Let’s consider the infinite thin transversely isotropic cylindrical shell. The elastic wave is spread along the axis 
that orthogonal of the plane of the isotropy. The transversely isotropic elastic medium is characterized by five elastic moduluses [1]: 
or by technical moduluses 
In the chosen orientation of the axis 
is the Joung’s modulus, 
is the shear modulus, 
is the Poisson’s ratio in the plane of the isotropy. 
and 
are the same values in the transverse plane. These moduluses connected with each other by the relationship [1-4]:
(1)
The Hooke’s law for the transversely isotropic elastic medium is written in the next form [1]:
(2)
where 
are components of the tensor of deformations, which are equals [1]:
(3)
where 
are components of the displacement vector 
Equations of the dynamic balance in the circular cylindrical system of coordinates [with the harmonic dependence from the time
] have the following appearance [1-4]:
(4)
where
(5)
Components of the displacement vector 
can be presented in the series form [2-4]:
(6)
where 
is the wave number of the elastic wave.
Then we substituted (4) in (5), we receive equations of the dynamic balance in displacements [2-4]:
(7)
(8)
(9)
where
Now if components of the displacement vector 
taken from (6) substitute in (7)-(9), then we receive following equations for radial functions 
[2-4]:
(10)
(11)
(12)
Boundary conditions: normal 
and tangent 
stresses are equal zero at external 
and internal 
surfaces of the elastic shell are added to equations (10)-(12) [2-4]:
(13)
(14)
(15)
where
3. Hypothesis of Thin Shells
The fellow parameter
can be used for thin shells, where
is middle radius and 
is the coordinate taking from the middle surface [2-5]:
(16)
We substitute decompositions in boundary Conditions (13)-(15) and 6 equations relative to 
unknown coefficients 
[2-4]:
(17)
(18)
(19)
(20)
(21)
(22)
The rest of equations can be received, by substitution of decompositions (16) in equations (10)-(12) and by equated of coefficients at identical powers 
[2-4]:
(23)
(24)
(25)
where 
.
It is necessary to use 
of equations (23)-(25) and for 
and 
coefficients with negative indexes are equal to zero. Then in common with the equations (17)-(22) the homogeneous system of 
equations relative to coefficients 
is formed. Afterwards, we expand the determinant of this system and let this determinant is equal zero we receive the characteristic equation for wave numbers 
of elastic waves of the mode 
in the transversely isotropic cylindrical shell.
Now we sell pay attention to elastic waves, which have axial symmetry: the dependence from the angle 
disappears. If vector of the shell displacement 
has not of the component
, then we have waves with the vertical polarization. In thin case components of strains 
and tangent stresses 
are equal to zero, but stresses 
and 
are equal [2-4]:
(26)
(27)
(28)
(29)
Equations of the dynamic balance (their only 2) have the following form [2-4]:
(30)
(31)
Displacements 
and 
can be taken in the form [2-4]:
(32)
(33)
For the thin shell 
and 
can be expanded in serieses:
(34)
(35)
Boundary conditions (their only 2) can be expressed as [2-4]:
(36)
(37)
The substitution (32), (33) and (34), (35) into boundary conditions (36), (37) and into equations of the dynamic balance (30), (31) results in the system of 
equations to calculate unknown coefficients 
The characteristic equation for wave numbers 
of elastic axisymmetrical waves in the transversely isotropic cylindrical shell we receive by expanding the determinant, which is equals zero. The axisymmetrical wave of the horizontal polarization (torsional wave) has only one component 
of the displacement vector 
The problem in this case has the analytic solution. Components of strains 
are equal to zero, but components of strains 
and 
are equal to:
The equation of the dynamic balance has the following form:
(38)
Used (2) and (3), we can describe (38) in the form:
(39)
The component Uφ can be presented as:
(40)
where 
is the torsional wave number.
We substitute (40) in (39) and have:
(41)
The Equation (41) is the Bessel’s equation for Bessel’s 
and Neiman 
functions of the first order:
(42)
where 
and 
are arbitrary constants;
From the boundary condition
, we receive the characteristic equation for torsional wave numbers
:
where
.
4. Conclusions
In the paper, we found the characteristic equation for wave numbers of elastic waves in thin transversely isotropic cylindrical shell with the help of the dynamic theory of the elasticity for the orthotropic medium and of the hypothesis of thin shells both for three—dimensional and axially symmetric problems.
5. Acknowledgments
The work was supported as part of research under State Contract no. P242 of April 21, 2010, within the Federal Target Program “Scientific and scientific—pedagogical personnel of innovative Russia for the 2009-2013”.